If \(Q_{\#}\)  is quadratic form on \(\mathbb {L}^{n/2}\)  polarized by the symmetric bilinear form \(F_{\#}\) , and \(T:\mathbb {L}\longrightarrow \mathbb {K},t\longmapsto t+t^{q}\)  is the trace form
from \(\mathbb {L}\)  onto \(\mathbb {K}\) , it follows that \(Q=T\circ Q_{\#}\)  is a
quadratic form on \(\mathbb {K}^{n}\)  polarized by the symmetric bilinear
form \(F=T\circ F_{\#}\) . As pointed out in [1]}, \( \Phi \)  maps singular 1-dimensional subspaces
of \(\mathbb {L}^{n/2}\)  onto totally singular 2-dimensional subspaces of \(\mathbb {K}^{n}\) , non-degenerate 1-dimensional subspaces of \(\mathbb {L}^{n/2} \)  onto non-degenerate 2-dimensional subspaces of \(\mathbb {K}^{n}\) 
and provides an embedding of \(O_{n/2}^{\varepsilon ^{\prime }}(q^{2})\)  in \(O_{n}^{\varepsilon }(q)\) . The approach in [1]} is geometric, its group-theoretical counterpart is contained in [3]}, Propositions 4.3.14, 4.3.16 and 4.3.20.
